CHAPTER 5 2014/IPPTChap005.pdf · production costs. 5-3 Chapter Overview . The Production Function • Mathematical function that defines the maximum amount of output that can be

CHAPTER 5

The Production Process and Costs

Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter Outline • The production function

– Short- versus long-run decisions – Measures of productivity – Manager’s role in production process – Algebraic forms of the production function and productivity – Isoquants and isocosts – Cost minimization and optimal input substitution

• The cost function – Short-run costs – Average and marginal costs – Relations among costs – Fixed and sunk costs – Algebraic forms of cost functions – Long-run costs and economies of scale

• Multiple-output cost functions – Economies of scope and cost complementarity

5-2

Chapter Overview

Introduction

• Chapter 4 focused on how consumers adjust consumption decisions in reaction to price and income changes. The theory developed illustrates the underlying principles of individual and market demand curves.

• This chapter examines how managers select the optimal mix of inputs that minimize production costs.

5-3

Chapter Overview

The Production Function

• Mathematical function that defines the maximum amount of output that can be produced with a given set of inputs.

𝑄𝑄 = 𝐹𝐹 𝐾𝐾, 𝐿𝐿 , where – 𝑄𝑄 is the level of output. –𝐾𝐾 is the quantity of capital input. – 𝐿𝐿 is the quantity of labor input.

5-4


Short-Run versus Long-Run Decisions: Fixed and Variable Inputs

• Short-run – Period of time where some factors of production

(inputs) are fixed, and constrain a manager’s decisions.

• Long-run – Period of time over which all factors of production

(inputs) are variable, and can be adjusted by a manager.

5-5


Measures of Productivity • Total product (TP)

– Maximum level of output that can be produced with a given amount of inputs.

• Average product (AP) – A measure of the output produced per unit of input.

• Average product of labor: 𝐴𝐴𝐴𝐴𝐿𝐿 = 𝑄𝑄𝐿𝐿

• Average product of capital: 𝐴𝐴𝐴𝐴𝐾𝐾 = 𝑄𝑄𝐾𝐾

• Marginal product (MP) – The change in total product (output) attributable to

the last unit of an input. • Marginal product of labor: 𝑀𝑀𝐴𝐴𝐿𝐿 = ∆𝑄𝑄

∆𝐿𝐿

• Marginal product of capital: 𝑀𝑀𝐴𝐴𝐾𝐾 = ∆𝑄𝑄∆𝐾𝐾

5-6


Measures of Productivity in Action

• Consider the following production function when 5 units of labor and 10 units of capital are combined produce: 𝑄𝑄 = 𝐹𝐹 10,5 = 150.

• Compute the average product of labor.

𝐴𝐴𝐴𝐴𝐿𝐿 = 1505

= 30 units per worker

• Compute the average product of capital.

𝐴𝐴𝐴𝐴𝐿𝐿 = 15010

= 15 units capital unit

5-7


Relation between Productivity Measures in Action

5-8

Labor input (holding capital constant)

0

Total product Average product Marginal product

Total product (TP)

Average product (APL)

Marginal product (MPL)

Increasing marginal returns to labor

Decreasing marginal returns to labor

Negative marginal returns to labor


The Manager’s Role in the Production Process

• Produce output on the production function. – Aligning incentives to induce maximum worker effort.

• Use the right mix of inputs to maximize profits. – To maximize profits when labor or capital vary in the

short run, the manager will hire: • Labor until the value of the marginal product of labor equals

the wage rate: 𝑉𝑉𝑀𝑀𝐴𝐴𝐿𝐿 = 𝑤𝑤, where 𝑉𝑉𝑀𝑀𝐴𝐴𝐿𝐿 = 𝐴𝐴 × 𝑀𝑀𝐴𝐴𝐿𝐿 • Capital until the value of the marginal product of capital

equals the rental rate: 𝑉𝑉𝑀𝑀𝐴𝐴𝐾𝐾 = 𝑟𝑟, where 𝑉𝑉𝑀𝑀𝐴𝐴𝐾𝐾 = 𝐴𝐴 × 𝑀𝑀𝐴𝐴𝐾𝐾

5-9


Manager’s Role in the Production Process in Action

• Suppose a firm sells its output in a competitive market where its output is sold at $5 per unit. If workers are also hired at a competitive wage of $200, what is the marginal productivity of the last worker?

• Since, 𝑉𝑉𝑀𝑀𝐴𝐴𝐿𝐿 = $5 × 𝑀𝑀𝐴𝐴𝐿𝐿 and 𝑤𝑤 = $200, then, $5 × 𝑀𝑀𝐴𝐴𝐿𝐿 = $200 ⇒ 𝑀𝑀𝐴𝐴𝐿𝐿 = 40 units

– The marginal productivity of the last unit of labor is 40 units.

– Alternatively, management should hire labor such that the last unit of labor produces 40 units.

5-10


Algebraic Forms of Production Functions

• Commonly used algebraic production function forms: – Linear: 𝑄𝑄 = 𝐹𝐹 𝐾𝐾, 𝐿𝐿 = 𝑎𝑎𝐾𝐾 + 𝑏𝑏𝐿𝐿, where 𝑎𝑎 and 𝑏𝑏 are constants.

– Leontief: 𝑄𝑄 = 𝐹𝐹 𝐾𝐾, 𝐿𝐿 = min 𝑎𝑎𝐾𝐾, 𝑏𝑏𝐿𝐿 , where 𝑎𝑎 and 𝑏𝑏 are constants.

– Cobb-Douglas: 𝑄𝑄 = 𝐹𝐹 𝐾𝐾, 𝐿𝐿 = 𝐾𝐾𝑎𝑎𝐿𝐿𝑏𝑏, where 𝑎𝑎 and 𝑏𝑏 are constants.

5-11


Algebraic Forms of Production Functions in Action

• Suppose that a firm’s estimated production function is:

𝑄𝑄 = 3𝐾𝐾 + 6𝐿𝐿 • How much output is produced when 3 units of

capital and 7 units of labor are employed? 𝑄𝑄 = 𝐹𝐹 3,7 = 3 3 + 6 7 = 51 units

5-12


Algebraic Measures of Productivity • Given the commonly used algebraic

production function forms, we can compute the measures of productivity as follows: – Linear:

• Marginal products: 𝑀𝑀𝐴𝐴𝐾𝐾 = 𝑎𝑎 and 𝑀𝑀𝐴𝐴𝐿𝐿 = 𝑏𝑏

• Average products: 𝐴𝐴𝐴𝐴𝐾𝐾 = 𝑎𝑎𝐾𝐾+𝑏𝑏𝐿𝐿𝐾𝐾

and 𝐴𝐴𝐴𝐴𝐿𝐿 = 𝑎𝑎𝐾𝐾+𝑏𝑏𝐿𝐿𝐿𝐿

– Cobb-Douglas: • Marginal products: 𝑀𝑀𝐴𝐴𝐾𝐾 = 𝑎𝑎𝐾𝐾𝑎𝑎−1𝐿𝐿𝑏𝑏and 𝑀𝑀𝐴𝐴𝐿𝐿 =𝑏𝑏𝐾𝐾𝑎𝑎−1𝐿𝐿𝑏𝑏

• Average products: 𝐴𝐴𝐴𝐴𝐾𝐾 = 𝐾𝐾𝑎𝑎𝐿𝐿𝑏𝑏

𝐾𝐾 and 𝐴𝐴𝐴𝐴𝐿𝐿 = 𝐾𝐾𝑎𝑎𝐿𝐿𝑏𝑏

𝐿𝐿

5-13


Algebraic Measures of Productivity in Action

• Suppose that a firm produces output according to the production function

𝑄𝑄 = 𝐹𝐹 1, 𝐿𝐿 = 1 1 4⁄ 𝐿𝐿3 4⁄ • Which is the fixed input?

– Capital is the fixed input.

• What is the marginal product of labor when 16 units of labor is hired?

𝑀𝑀𝐴𝐴𝐿𝐿 = 1 ×34𝐿𝐿−

14 = 1 ×

34

16 −14 =38

5-14


Isoquants and Marginal Rate of Technical Substitution

• Isoquants capture the tradeoff between combinations of inputs that yield the same output in the long run, when all inputs are variable.

• Marginal rate of technical substitutions (MRTS) – The rate at which a producer can substitute between

two inputs and maintain the same level of output. – Absolute value of the slope of the isoquant.

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 =𝑀𝑀𝐴𝐴𝐿𝐿𝑀𝑀𝐴𝐴𝐾𝐾

5-15


Isoquants and Marginal Rate of Technical Substitution in Action

5-16

Labor Input 0

A

B

𝑄𝑄𝐼𝐼=100 units of output

𝑄𝑄2 =200 units of output

𝑄𝑄3 = 300 units of output

Capital Input


Diminishing Marginal Rate of Technical Substitution in Action

5-17

Labor Input (L) 0

D

C

𝑄𝑄0=100 units

Capital Input (K)

B A

∆𝐿𝐿 = −1 ∆𝐿𝐿 = −1

∆𝐾𝐾 =3

∆𝐾𝐾 = 1

Slope (at C): ∆𝐾𝐾∆𝐿𝐿

= −31

= −3 = −𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

Slope (at A): ∆𝐾𝐾∆𝐿𝐿

= −11



Isocost and Changes in Isocost Lines

• Isocost – Combination of inputs that yield cost the same cost.

𝑤𝑤𝐿𝐿 + 𝑟𝑟𝐾𝐾 = 𝐶𝐶 or, re-arranging to the intercept-slope formulation:

𝐾𝐾 =𝐶𝐶𝑟𝑟−𝑤𝑤𝑟𝑟𝐿𝐿

• Changes in isocosts – For given input prices, isocosts farther from the origin

are associated with higher costs. – Changes in input prices change the slopes of isocost

lines.

5-18


Isocost Line

5-19

Labor Input (L) 0

Capital Input (K)

𝐿𝐿

𝐶𝐶𝑟𝑟


𝐶𝐶𝑤𝑤

𝐾𝐾 =𝐶𝐶𝑟𝑟 −

𝑤𝑤𝑟𝑟 𝐿𝐿

Changes in the Isocost Line

5-20

Labor Input (L) 0

Capital Input (K)

𝐶𝐶0

𝑟𝑟


Less expensive input bundles

𝐶𝐶0

𝑤𝑤 𝐶𝐶1

𝑤𝑤

𝐶𝐶1

𝑟𝑟

More expensive input bundles

Changes in the Isocost Line

5-21

Labor Input (L) 0

Capital Input (K)


𝐶𝐶𝑤𝑤1

𝐶𝐶𝑤𝑤0

𝐶𝐶𝑟𝑟

Due to increase in wage rate 𝑤𝑤1 > 𝑤𝑤0

Cost-Minimization Input Rule in Action

5-22

Labor Input (L) 0

𝑄𝑄𝐼𝐼=100 units

Capital Input (K)

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 =𝑤𝑤𝑟𝑟


𝐶𝐶2

𝑟𝑟

𝐶𝐶2

𝑤𝑤 𝐶𝐶1

𝑤𝑤

𝐶𝐶1

𝑟𝑟

𝐴𝐴

Cost Minimization and the Cost-Minimizing Input Rule

• Cost minimization – Producing at the lowest possible cost.

• Cost-minimizing input rule – Produce at a given level of output where the marginal

product per dollar spent is equal for all inputs: 𝑀𝑀𝐴𝐴𝐿𝐿𝑤𝑤

=𝑀𝑀𝐴𝐴𝐾𝐾𝑟𝑟

– Equivalently, a firm should employ inputs such that the marginal rate of technical substitution equals the ratio of input prices:

𝑀𝑀𝐴𝐴𝐿𝐿𝑀𝑀𝐴𝐴𝐾𝐾

=𝑤𝑤𝑟𝑟

5-23


Cost-Minimizing Input Rule in Action

• Suppose that labor and capital are hired at a competitive wage of $10 and $25, respectively. If the marginal product of capital is 6 units and the marginal product of labor is 3 units, is the firm hiring the cost-minimizing units of capital and labor? – Since 3

$10> 6

$25⇒ 𝑀𝑀𝑀𝑀𝐿𝐿

𝑤𝑤> 𝑀𝑀𝑀𝑀𝐾𝐾

𝑟𝑟, the marginal

product per dollar spent on labor exceeds the marginal product per dollar spent on capital.

– The firm is not minimizing costs and should use fewer units of capital and more labor.

5-24


Optimal Input Substitution in Action

5-25

Labor Input (L) 0

B

𝑄𝑄0

Capital Input (K)

New cost-minimizing point due to higher wage

𝐿𝐿1

𝐾𝐾2

𝐿𝐿2

𝐾𝐾1 A

Initial point of cost minimization


H

I

F

J

G

The Cost Function • Mathematical relationship that relates cost to the

cost-minimizing output associated with an isoquant.

• Short-run costs – Fixed costs: 𝐹𝐹𝐶𝐶 – Sunk costs – Short-run variable costs: 𝑉𝑉𝐶𝐶 𝑄𝑄 – Short-run total costs: 𝑀𝑀𝐶𝐶 𝑄𝑄 = 𝐹𝐹𝐶𝐶 + 𝑉𝑉𝐶𝐶 𝑄𝑄

• Long-run costs – All costs are variable – No fixed costs

5-26

The Cost Function

Short-Run Costs in Action

5-27

Output 0

𝑀𝑀𝐶𝐶 𝑄𝑄 = 𝐹𝐹𝐶𝐶 + 𝑉𝑉𝐶𝐶 𝑄𝑄

𝑉𝑉𝐶𝐶 𝑄𝑄

𝐹𝐹𝐶𝐶

𝐹𝐹𝐶𝐶

Total costs Variable costs Fixed costs

𝐹𝐹𝐶𝐶

The Cost Function

Average and Marginal Costs • Average costs

– Average fixed: 𝐴𝐴𝐹𝐹𝐶𝐶 = 𝐹𝐹𝐹𝐹𝑄𝑄

– Average variable costs: 𝐴𝐴𝑉𝑉𝐶𝐶 = 𝑉𝑉𝐹𝐹 𝑄𝑄𝑄𝑄

– Average total cost: 𝐴𝐴𝑀𝑀𝐶𝐶 = 𝐹𝐹 𝑄𝑄𝑄𝑄

• Marginal cost – The (incremental) cost of producing an additional

unit of output.

– 𝑀𝑀𝐶𝐶 = ∆𝐹𝐹∆𝑄𝑄

5-28

The Cost Function

The Relationship between Average and Marginal Costs in Action

5-29

Output 0

𝐴𝐴𝑀𝑀𝐶𝐶

A𝑉𝑉𝐶𝐶 𝑀𝑀𝐶𝐶

ATC, AVC, AFC and MC ($)

𝐴𝐴𝐹𝐹𝐶𝐶

Minimum of ATC

Minimum of AVC

The Cost Function

Fixed and Sunk Costs

• Fixed costs – Cost that does not change with output.

• Sunk cost – Cost that is forever lost after it has been paid.

• Principle of Irrelevance of Sunk Costs

– A decision maker should ignore sunk costs to maximize profits or minimize loses.

5-30

The Cost Function

Long-Run Costs

• In the long run, all costs are variable since a manager is free to adjust levels of all inputs.

• Long-run average cost curve – A curve that defines the minimum average cost of

producing alternative levels of output, allowing for optimal selection of both fixed and variable factors of production.

5-31

The Cost Function

Long-Run Average Total Costs in Action

5-32

Output 0

LRAC ($)

𝐴𝐴𝑀𝑀𝐶𝐶0

𝐴𝐴𝑀𝑀𝐶𝐶1

𝐴𝐴𝑀𝑀𝐶𝐶2 𝐿𝐿𝑀𝑀𝐴𝐴𝐶𝐶

The Cost Function

𝑄𝑄∗

Economies of Scale • Economies of scale

– Portion of the long-run average cost curve where long-run average costs decline as output increases.

• Diseconomies of scale – Portion of the long-run average cost curve where

long-run average costs increase as output increases.

• Constant returns to scale – Portion of the long-run average cost curve that

remains constant as output increases.

5-33

The Cost Function

Economies and Diseconomies of Scale in Action

5-34

Output 0

LRAC ($)

𝐿𝐿𝑀𝑀𝐴𝐴𝐶𝐶

The Cost Function

Economies of scale Diseconomies of scale

𝑄𝑄∗

Constant Returns to Scale in Action

5-35

Output 0

LRAC ($)

𝐴𝐴𝑀𝑀𝐶𝐶1 𝐴𝐴𝑀𝑀𝐶𝐶2 𝐴𝐴𝑀𝑀𝐶𝐶3

𝐿𝐿𝑀𝑀𝐴𝐴𝐶𝐶

The Cost Function

Multiple-Output Cost Function

• Economies of scope – Exist when the total cost of producing 𝑄𝑄1 and 𝑄𝑄2

together is less than the total cost of producing each of the type of output separately.

𝐶𝐶 𝑄𝑄1, 0 + 𝐶𝐶 0,𝑄𝑄2 > 𝐶𝐶 𝑄𝑄1,𝑄𝑄2

• Cost complementarity – Exist when the marginal cost of producing one

type of output decreases when the output of another good is increased.

∆𝑀𝑀𝐶𝐶1 𝑄𝑄1,𝑄𝑄2∆𝑄𝑄2

< 0

5-36


Multiple-Output Cost Function in Action • Suppose a firm produces two goods and has cost

function given by 𝐶𝐶 = 100 − 0.5𝑄𝑄1𝑄𝑄2 + 𝑄𝑄1 2 + 𝑄𝑄2 2

• If the firm plans to produce 4 units of 𝑄𝑄1 and 6 units of 𝑄𝑄2 – Does this cost function exhibit cost

complementarities? • Yes, cost complementarities exist since

𝑎𝑎 = −0.5 < 0 – Does this cost function exhibit economies of scope?

• Yes, economies of scope exist since 100− 0.5 4 6 > 0

5-37


Conclusion

• To maximize profits (minimize costs) managers must use inputs such that the value of marginal product of each input reflects the price the firm must pay to employ the input.

• The optimal mix of inputs is achieved when the 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 = 𝑤𝑤

𝑟𝑟.

• Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters.

5-38

5-39

Labor Input (L) 0


Capital Input (K)

𝐿𝐿

𝐾𝐾

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 =𝑤𝑤𝑟𝑟


Cost-Minimization In Action

Diminishing Marginal Rate of Technical Substitution in Action

5-40

Labor Input (L) 0

A

B


Capital Input (K)

C D

∆𝐿𝐿 = 1 ∆𝐿𝐿 = 1

∆𝐾𝐾 = −5

∆𝐾𝐾 = −1

Slope (at A): ∆𝐾𝐾∆𝐿𝐿

= −51


Slope (at C): ∆𝐾𝐾∆𝐿𝐿

= −11



• Suppose that capital and labor are hired at a competitive wage of $10 and $20, respectively. If the marginal product of capital is 5 units and the marginal product of labor is 10 units, is the firm hiring the cost-minimizing units of capital and labor?

– Since 5$10

= 10$20

⇒ 𝑀𝑀𝑀𝑀𝐾𝐾𝑟𝑟

= 𝑀𝑀𝑀𝑀𝐿𝐿𝑤𝑤

, the cost-minimizing mix of capital and labor is being utilized.

5-41


Cost-Minimization Input Rule In Action

Multiple-Output Cost Function in Action • Suppose a firm produces two goods and has cost

function given by 𝐶𝐶 = 500 − 0.3𝑄𝑄1𝑄𝑄2 + 𝑄𝑄1 2 + 𝑄𝑄2 2

• If the firm plans to produce 4 units of 𝑄𝑄1 and 6 units of 𝑄𝑄2 – Does this cost function exhibit cost

complementarities? • Yes, cost complementarities exist since

𝑎𝑎 = −0.3 < 0 – Does this cost function exhibit economies of scope?

• Yes, economies of scope exist since 500− 0.3 4 6 > 0

5-42


Documents

CHAPTER 5 2014/IPPTChap005.pdf · production costs. 5-3 Chapter Overview . The Production Function • Mathematical function that defines the maximum amount of output that can be